The holonomy inverse problem

Abstract

Let (M,g) be a smooth Anosov Riemannian manifold and \mathcal{C}^\sharp the set of its primitive closed geodesics. Given a Hermitian vector bundle \mathcal{E} equipped with a unitary connection \nabla^{\mathcal{E}} , we define \mathcal{T}^\sharp(\mathcal{E}, \nabla^{\mathcal{E}}) as the sequence of traces of holonomies of \nabla^{\mathcal{E}} along elements of \mathcal{C}^\sharp . This descends to a homomorphism on the additive moduli space \mathbb{A} of connections up to gauge \mathcal{T}^\sharp: (\mathbb{A}, \oplus) \to \ell^\infty(\mathcal{C}^\sharp) , which we call the primitive trace map . It is the restriction of the well-known Wilson loop operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map \mathcal{T}^\sharp is locally injective near generic points of \mathbb{A} when \dim(M) \geq 3 . We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of \mathbb{A} to the Pollicott–Ruelle resonance near zero of a certain natural transport operator.